Is Blender 2.81 able to pack Animated Textures?
From the Blender Manual:

Not all external files can be packed
Some typically heavy external files, like video ones from the Sequence Editor or Movie Clips, cannot be packed in a blend-file.

Consider what would happen if you did pack those files in. Every time Blender makes a backup save (which it does by default), it'd have to make a copy of all the data, including the video files. If you're considering uploading your project to a render farm, some allow for video files to be sent in an accompanying zip file.
If you're just transferring the file from computer to computer, consider using Relative Paths and keeping all your files in organized folders.